symmetryS S

Article

Dual Hesitant q-Rung Orthopair Fuzzy HamacherAggregation Operators and their Applications inScheme Selection of Construction Project

Ping Wang 1, Guiwu Wei 2 , Jie Wang 2, Rui Lin 3 and Yu Wei 4,*1 School of Engineering, Sichuan Normal University, Chengdu 610101, China; WangPing97@163.com2 School of Business, Sichuan Normal University, Chengdu 610101, China;

weiguiwu1973@sicnu.edu.cn (G.W.); JW970326@163.com (J.W.)3 School of Economics and Management, Chongqing University of Arts and Sciences, Chongqing 402160,

China; linrui@cqwu.edu.cn4 School of Finance, Yunnan University of Finance and Economics, Kunming 650221, China* Correspondence: weiyusy@126.com

Received: 6 May 2019; Accepted: 30 May 2019; Published: 6 June 2019�����������������

Abstract: The q-rung orthopair fuzzy set (q-ROFS), which is the extension of intuitionistic fuzzyset (IFS) and Pythagorean fuzzy set (PFS), satisfies the sum of q-th power of membership degreeand nonmembership degree is limited 1. Evidently, the q-ROFS can depict more fuzzy assessmentinformation and consider decision-maker’s (DM’s) hesitance. Thus, the concept of a dual hesitantq-rung orthopair fuzzy set (DHq-ROFS) is developed in this paper. Then, based on Hamacheroperation laws, weighting average (WA) operator and weighting geometric (WG) operator, some dualhesitant q-rung orthopair fuzzy Hamacher aggregation operators are developed, such as the dualhesitant q-rung orthopair fuzzy Hamacher weighting average (DHq-ROFHWA) operator, the dualhesitant q-rung orthopair fuzzy Hamacher weighting geometric (DHq-ROFHWG) operator, the dualhesitant q-rung orthopair fuzzy Hamacher ordered weighted average (DHq-ROFHOWA) operator,the dual hesitant q-rung orthopair fuzzy Hamacher ordered weighting geometric (DHq-ROFHOWG)operator, the dual hesitant q-rung orthopair fuzzy Hamacher hybrid average (DHq-ROFHHA)operator, and the dual hesitant q-rung orthopair fuzzy Hamacher hybrid geometric (DHq-ROFHHG)operator. The precious merits and some particular cases of above mentioned aggregation operatorsare briefly introduced. In the end, an actual application for scheme selection of construction project isprovided to testify the proposed operators and deliver a comparative analysis.

Keywords: multiple attribute decision-making (MADM) problems; Hamacher operation laws; dualhesitant q-rung orthopair fuzzy set (DHq-ROFS); the DHq-ROFHWA operator; the DHq-ROFHWGoperator

1. Introduction

In real-life decision-making problems, how to select the most desirable alternative from a givenalternative set is very important. The most common method is fusing the evaluation informationgiven by experts, and ranking all alternatives according to fused results to select best one(s). Thus,how to derive reasonable evaluation information is worth studying. To do this, Atanassov [1] firstlyextended the fuzzy set (FS) [2] and introduced intuitionistic fuzzy set (IFS). The intuitionistic fuzzy set(IFS) is mainly characterized by the function of membership degree µ and nonmembership degree v,which satisfies µ+ v ≤ 1. The intuitionistic fuzzy set (IFS) and its extensions have attracted a largeamount of scholars’ attention since its emergence [3–16]. More recently, the Pythagorean fuzzy set(PFS) [17] has been proposed to depict more fuzzy assessment information. The PFS is also consisted

Symmetry 2019, 11, 771; doi:10.3390/sym11060771 www.mdpi.com/journal/symmetry

Symmetry 2019, 11, 771 2 of 26

of the membership degree µ and nonmembership degree v, which satisfies µ2 + v2≤ 1, so, it is obvious

that the PFS can express more assessment information than the IFS. However, the scope of assessmentinformation is still limited under Pythagorean fuzzy environment. For instance, given the evaluationvalue (0.7,0.9), we can easily find that 0.72 + 0.92

≤ 1, which indicates that PFS cannot deal withsuch MADM problems. Then, to describe more evaluation information, Yager [18] further definedthe q-rung orthopair fuzzy set (q-ROFS), q-ROFS is also consisted of the membership degree µ andnonmembership degree v which satisfies µq + vq

≤ 1. Obviously, q-ROFS can be regarded as theextension of the IFS and PFS, when q = 1, the q-ROFS reduces to IFS, when q = 2, the q-ROFS reducesto PFS (see Figure 1). Afterwards, more and more works about q-ROFS have been studied by numerousscholars [19–25].

Symmetry 2019, 11, x FOR PEER REVIEW 2 of 29

The PFS is also consisted of the membership degree and nonmembership degree v , which

satisfies 2 2 1v + , so, it is obvious that the PFS can express more assessment information than

the IFS. However, the scope of assessment information is still limited under Pythagorean fuzzy

environment. For instance, given the evaluation value (0.7,0.9), we can easily find that 2 20.7 0.9 1,+ which indicates that PFS cannot deal with such MADM problems. Then, to

describe more evaluation information, Yager [18] further defined the q-rung orthopair fuzzy set

(q-ROFS), q-ROFS is also consisted of the membership degree and nonmembership degree v

which satisfies 1.q qv + Obviously, q-ROFS can be regarded as the extension of the IFS and

PFS, when 1q = , the q-ROFS reduces to IFS, when 2q = , the q-ROFS reduces to PFS.

Afterwards, more and more works about q-ROFS have been studied by numerous scholars [19–25].

Figure 1. The relationship between intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PFS), and

q-rung orthopair fuzzy set (q-ROFS).

However, the above-mentioned methods do not consider the human’s hesitance. In order to

overcome this limitation, the hesitant fuzzy sets (HFSs) [26] and dual hesitant fuzzy sets (DHFSs)

[27,28] have been proposed to deal with MADM issues effectively. Combining the advantages of the

two fuzzy sets, Xu et al. [29] gave the concept of the dual hesitant q-rung orthopair fuzzy set

(DHq-ROFS) and presented some Heronian mean operators for MADM. Obviously, the dual

hesitant q-rung orthopair fuzzy numbers (DHq-ROFNs) can express evaluation information more

convenience in actual MADM applications.

In the decision-making process, the way to express evaluation information is only one aspect;

another vital aspect is fusing this information. Hamacher operations [30], which include Hamacher

product and Hamacher sum, can replace the traditional algebraic product and algebraic sum,

respectively. In past few years, numerous investigators studied the Hamacher aggregation operators

and their applications [31–41]. In this paper, based on Hamacher operations, we shall develop some

new operation laws of DHq-ROFNs, then, by utilizing the new operation laws, we can aggregate

dual hesitant q-rung orthopair fuzzy information by Hamacher WA and Hamacher WG operator.

The Hamacher operations have the advantage of considering the relationship between the values

being fused, thus the fused results are more reasonable and accuracy. Clearly, DHq-ROFN is a

meaningful tool to express evaluation information; Hamacher operations are good to fuse evaluation

information, so it’s worth to develop some Hamacher operators under dual hesitant q-rung

orthopair fuzzy environments.

The mainly novelty and contribution of our manuscript is developing some new Hamacher

operators to aggregate the dual hesitant q-rung orthopair fuzzy information. Evidently, these

operators have the following advantages. (1) The DHq-ROFS can not only extend the scope of the

assessment information to depict more fuzzy information, but also consider the human’s hesitance,

thus it is more useful and reasonable to derive decision-making results. (2) The Hamacher

operations can consider the relationship between fused arguments, obviously, Hamacher operations

are more suitable for handling practical MADM problems. Thus, it is of great significance to propose

some new operators based on the dual hesitant q-rung orthopair fuzzy information and Hamacher

operations.

Figure 1. The relationship between intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PFS), and q-rungorthopair fuzzy set (q-ROFS).

However, the above-mentioned methods do not consider the human’s hesitance. In order toovercome this limitation, the hesitant fuzzy sets (HFSs) [26] and dual hesitant fuzzy sets (DHFSs) [27,28]have been proposed to deal with MADM issues effectively. Combining the advantages of the twofuzzy sets, Xu et al. [29] gave the concept of the dual hesitant q-rung orthopair fuzzy set (DHq-ROFS)and presented some Heronian mean operators for MADM. Obviously, the dual hesitant q-rungorthopair fuzzy numbers (DHq-ROFNs) can express evaluation information more convenience inactual MADM applications.

In the decision-making process, the way to express evaluation information is only one aspect;another vital aspect is fusing this information. Hamacher operations [30], which include Hamacherproduct and Hamacher sum, can replace the traditional algebraic product and algebraic sum,respectively. In past few years, numerous investigators studied the Hamacher aggregation operatorsand their applications [31–41]. In this paper, based on Hamacher operations, we shall develop somenew operation laws of DHq-ROFNs, then, by utilizing the new operation laws, we can aggregatedual hesitant q-rung orthopair fuzzy information by Hamacher WA and Hamacher WG operator.The Hamacher operations have the advantage of considering the relationship between the values beingfused, thus the fused results are more reasonable and accuracy. Clearly, DHq-ROFN is a meaningful toolto express evaluation information; Hamacher operations are good to fuse evaluation information, so it’sworth to develop some Hamacher operators under dual hesitant q-rung orthopair fuzzy environments.

The mainly novelty and contribution of our manuscript is developing some new Hamacheroperators to aggregate the dual hesitant q-rung orthopair fuzzy information. Evidently, these operatorshave the following advantages. (1) The DHq-ROFS can not only extend the scope of the assessmentinformation to depict more fuzzy information, but also consider the human’s hesitance, thus it is moreuseful and reasonable to derive decision-making results. (2) The Hamacher operations can consider therelationship between fused arguments, obviously, Hamacher operations are more suitable for handlingpractical MADM problems. Thus, it is of great significance to propose some new operators based onthe dual hesitant q-rung orthopair fuzzy information and Hamacher operations.

To achieve this goal, the rest of our article is constructed as follows. Section 2 introduces someworks on Pythagorean fuzzy set and q-rung orthopair fuzzy set. Section 3 briefly reviews some

Symmetry 2019, 11, 771 3 of 26

fundamental theories of q-ROFSs and DHq-ROFSs. Section 4 introduces some HWA and HWGoperators under DHq-ROFS environment, such as the DHq-ROFHWA operator, the DHq-ROFHWGoperator, the DHq-ROFHOWA operator, the DHq-ROFHOWG operator, the DHq-ROFHHA operator,and the (DHq-ROFHHG operator. Section 5 proposes an actual application for scheme selection ofconstruction project with DHq-ROFNs and compares our developed operators with other existingmethods in this filed. Section 6 concludes the paper with some remarks.

2. Literature Review

In previous literature, research on the Pythagorean fuzzy set (PFS) has been conducted by manyscholars. Zhang and Xu [42] defined the Pythagorean fuzzy TOPSIS model to solve the MADMproblems. Peng and Yang [43] primarily proposed two Pythagorean fuzzy operations, includingthe division and subtraction operations, to better understand PFS. Reformat and Yager [44] handledthe collaborative-based recommender system with Pythagorean fuzzy information. Combined theMaclaurin Symmetric Mean (MSM) [45] operators and Pythagorean fuzzy information, Yang andPang [46] developed some new Pythagorean fuzzy interaction MSM operators to handle MADMproblems. Gou et al. [47] studied some precious properties of continuous Pythagorean fuzzy assessmentinformation. Yang et al. [48] studied the partitioned Bonferroni mean (PBM) operators underPythagorean fuzzy environment and defined some Pythagorean fuzzy interaction PBM operators tosolve MADM. Based on Hamacher operation laws and Pythagorean fuzzy information, Wu and Wei [49]proposed some new aggregation operators to fuse Pythagorean fuzzy information and applied them toMADM problems. Liang et al. [50] studied the Pythagorean fuzzy set (PFS) based on the GA operationsand Bonferroni mean (BM) operators. Ren et al. [51] developed the Pythagorean fuzzy TODIM model.Wei and Lu [52] developed Pythagorean fuzzy MSM (PFMSM) operator and Pythagorean fuzzyweighted MSM (PFWMSM) operator for MADM. Liang et al. [53] defined some novel Bonferroni meanoperators under PFS environment. Consider the interrelationship between being fused arguments,Li et al. [11] gave some new Pythagorean fuzzy aggregation operators for selection of green supplierbased on traditional Hamy mean (HM) operators. Peng et al. [54] presented some novel Pythagoreanfuzzy information measures for MADM problems. On account of the PFSs [17,55] and DHFSs [27,28],Xu and Wei [56] further defined the dual hesitant Pythagorean fuzzy sets (DHPFSs), then, based onHamacher operation laws weighting average (WA) operator and weighting geometric (WG) operator,some new aggregation operators under dual hesitant Pythagorean fuzzy environment were developedfor MADM problems.

In terms of the q-ROFS, according to the traditional WA and WG operators, Liu and Wang [57]introduced two q-rung orthopair fuzzy aggregation operators to fuse q-rung orthopair fuzzy numbers(q-ROFNs). Combined q-rung orthopair fuzzy information and MSM operators, Wei et al. [58] proposedsome new q-rung orthopair fuzzy aggregation operators. Bai et al. [59] defined some q-rung orthopairfuzzy Partitioned Maclaurin Symmetric Mean (q-ROFPMSM) operators for MADM. Liu et al. [60]developed some q-rung orthopair fuzzy Power MSM operators. Liu et al. [61] developed someextended Bonferroni mean operators under q-rung orthopair fuzzy environment. Liu and Liu [62]presented some Bonferroni mean operators to fuse q-rung orthopair fuzzy information; Liu and Liu [63]proposed the concept of linguistic q-rung orthopair fuzzy set (Lq-ROFS) and introduced some PBMoperators to fuse linguistic q-rung orthopair fuzzy information. Yang and Pang [64] studied PartitionedBonferroni mean operators under q-rung orthopair fuzzy environment. The contribution of differentauthors under q-ROFNs is listed in Table 1.

Wei et al. [65] defined some q-rung orthopair fuzzy Heronian mean (q-ROFHM) operator.Liu et al. [66] provided some Heronian mean operator to aggregate q-ROFNs. In this paper, accordingto the dual hesitant q-rung orthopair set defined by Xu et al. [29], we shall propose some q-rungorthopair fuzzy Hamacher operation laws to fuse the q-rung orthopair fuzzy information. The goal ofour paper is to develop some operators that can consider human’s hesitance and the interrelationshipbetween being fused arguments.

Symmetry 2019, 11, 771 4 of 26

Table 1. The contribution of different authors under q-ROFNs.

Authors Production Consider theInterrelationship

Consider theParameter

Vector

Consider theHuman’sHesitancy

Consider theOrder Position

Weights andItself Weights

Liu and Wang [57] q-ROFWA operator No No No NoLiu and Wang [57] q-ROFWG operator No No No No

Wei, et al. [58] q-ROFMSM operators Yes Yes No No

Bai, et al. [59] q-ROF-Partitioned-MSMoperators Yes Yes No No

Liu, et al. [60] q-ROF-Power-MSMoperators Yes Yes No No

Liu, et al. [61] q-ROFEBM operators Yes Yes No NoLiu and Liu [62] q-ROFBM operators Yes Yes No No

Liu and Liu [63] Lq-ROF-Power-BMoperators Yes Yes No No

Yang and Pang [64] q-ROF-Partitioned-BMoperators Yes Yes No No

Wei, et al. [65] q-R2TLOFHM operators Yes Yes No NoLiu, et al. [66] q-ROFHM operators Yes Yes No NoXu, et al. [29] q-RDHOFHM operators Yes Yes Yes No

Proposed model DHq-ROFHHA andDHq-ROFHHG operators Yes Yes Yes Yes

3. Preliminaries

3.1. The q-Rung Orthopair Fuzzy Set

As the generalization of IFS and PFS, the basic definition, score function, accuracy function,and operation laws of the q-rung orthopair fuzzy sets (q-ROFSs) [18] can be listed as below.

Definition 1 [18]. Let X be a fix set. A q-rung orthopair fuzzy set can be denoted as

P ={⟨

x, (µP(x), νP(x))⟩|x ∈ X

}(1)

where µP : X→ [0, 1] indicates the function of membership degree and νP : X→ [0, 1] indicates the functionof nonmembership degree, which satisfies(

µp(x))q+

(νp(x)

)q≤ 1, q ≥ 1 (2)

Based on membership degree and nonmembership degree, the indeterminacy degree can be calculated as

πp(x) =q√(µp(x)

)q+

(νp(x)

)q−

(µp(x)

)q(νp(x)

)q(3)

For convenience, we named p = (µ, ν) a q-rung orthopair fuzzy number (q-ROFN).

Definition 2 [57]. Suppose that p1 = (µ1, ν1) and p2 = (µ2, ν2) be two q-ROFNs, let s(p1) =12

(1 + (µ1)

q− (ν1)

q)

and s(p2) = 12

(1 + (µ2)

q− (ν2)

q)

be the score results of p1 and p2, let H(p1) =

(µ1)q + (ν1)

q and H(p2) = (µ2)q + (ν2)

q be the accuracy results of p1 and p2, then we can give the comparativelaws between any two q-ROFNs: if s(p1) < s(p2), then p1 < p2; if s(p1) = s(p2), then (1) if H(p1) = H(p2),then p1 = p2; (2) if H(p1) < H(p2), p1 < p2.

Symmetry 2019, 11, 771 5 of 26

Definition 3 [57]. Assume that p1 = (µ1, ν1), p2 = (µ2, ν2), and p = (µ, ν) be three q-ROFNs, then

(1) p1 ⊕ p2 =(

q√(µ1)

q + (µ2)q− (µ1)

q(µ2)q, ν1ν2

);

(2) p1 ⊗ p2 =(µ1µ2, q

√(ν1)

q + (ν2)q− (ν1)

q(ν2)q);

(3) λp =

(q√

1− (1− µq)λ, νλ),λ > 0;

(4) (p)λ =

(µλ,

q√

1− (1− νq)λ),λ > 0;

(5) pc = (ν,µ).

3.2. Dual Hesitant q-Rung Orthopair Fuzzy Set

In accordance of the q-ROFSs and DHFSs [27,28], we further introduce the dual hesitant q-rungorthopair fuzzy sets (DHq-ROFSs) [29] as follows.

Definition 4 [29]. Let X be a fix set, then a DHq-ROFS on X can be denoted as

d = (⟨x, hP(x), gP(x)

⟩|x ∈ X ) (4)

where hP(x) indicates membership hesitancy set with several values in [0,1], gP(x) indicates nonmembershiphesitancy set with values in [0, 1], which satisfies

∪α∈h (max(α))q + ∪β∈g(max(β))q≤ 1 (5)

where α ∈ hP(x), β ∈ gP(x). Then we named d(x) = (hP(x), gP(x)) a dual hesitant q-rung orthopairfuzzy number (DHq-ROFN) described by d = (h, g), which satisfies α ∈ h, β ∈ g, 0 ≤ α, β ≤ 1 and∪α∈h(max(α))q + ∪β∈g(max(β))q

≤ 1.

Definition 5 [29]. Assume that d = (h, g) is a DHq-ROFN, let s(d) = 12

(1 + 1

#h∑α∈h α

q−

1#g

∑β∈g β

q)

be the

score results of d = (h, g) and E(d) = 1#h

∑α∈h α

q + 1#g

∑β∈g β

q be the accuracy results of d = (h, g), where #hindicates the number of elements in set h and #g indicates the number of elements in set g, respectively, assumethat di = (hi, gi)(i = 1, 2) be any two DHq-ROFNs, then if s(d1) > s(d2), then d1 � d2; if s(d1) = s(d2),then: (1) If E(d1) = E(d2), then d1 = d2; (2) If E(d1) > E(d2), then d1 � d2.

Definition 6 [29]. Let d1 = (h1, g1), d2 = (h2, g2), and d = (h, g) be three DHq-ROFNs, then, some newoperations on the DHq-ROFNs are defined as

(1) dλ = ∪α∈h,β∈g

{{αλ

},{

q√

1− (1− βq)λ}}

,λ > 0;

(2) λd = ∪α∈h,β∈g

{{q√

1− (1− αq)λ}

,{βλ

}},λ > 0;

(3) d1 ⊕ d2 = ∪α1∈h1,α2∈h2,β1∈g1,β2∈g2

{{q√(α1)

q + (α2)q− (α1)

q(α2)q},{β1β2

}};

(4) d1 ⊗ d2 = ∪α1∈h1,α2∈h2,β1∈g1,β2∈g2

{{α1α2},

{q√(β1)

q + (β2)q− (β1)

q(β2)q}}

.

3.3. Hamacher Operations of Dual Hesitant q-rung Orthopair Fuzzy Set

Definition 7. Let d1 = (h1, g1), d2 = (h2, g2), and d = (h, g) be three DHq-ROFNs, γ > 0, and based on thetraditional Hamacher operations [30], some basic Hamacher operations of DHq-ROFNS are defined as follows

Symmetry 2019, 11, 771 6 of 26

d1 ⊕ d2 = ∪α1∈h1,α2∈h2,β1∈g1,β2∈g2

({q

√(α1)

q+(α2)q−(α1)

q(α2)q−(1−γ)(α1)

q(α2)q

1−(1−γ)(α1)q(α2)

q

}, β1β2

q√γ+(1−γ)((β1)

q+(β2)q−(β1)

q(β2)q)

;

(6)

d1 ⊗ d2 = ∪α1∈h1,α2∈h2,β1∈g1,β2∈g2

α1α2

q√γ+(1−γ)((α1)

q+(α2)q−(α1)

q(α2)q)

,{q

√(β1)

q+(β2)q−(β1)

q(β2)q−(1−γ)(β1)

q(β2)q

1−(1−γ)(β1)q(β2)

q

});

(7)

λd = ∪α∈h,β∈g

q

√(1+(γ−1)(α)q)

λ−(1−(α)q)

λ

(1+(γ−1)(α)q)λ+(γ−1)(1−(α)q)

λ

q√γ(β)λ

q√(1+(γ−1)(1−(β)q))

λ+(γ−1)(β)qλ

;

(8)

dλ = ∪α∈h,β∈g

q

√(1+(γ−1)(β)q)

λ−(1−(β)q)

λ

(1+(γ−1)(β)q)λ+(γ−1)(1−(β)q)

λ

q√γ(α)λ

q√(1+(γ−1)(1−(α)q))

λ+(γ−1)(α)qλ

.

(9)

4. Dual Hesitant q-Rung Orthopair Fuzzy Hamacher Operators

4.1. Dual Hesitant q-Rung Orthopair Fuzzy Hamacher Averaging Operators

In this section, based on the Hamacher operations of dual hesitant q-rung orthopair fuzzy numbers(DHq-ROFNs), we shall present some dual hesitant q-rung orthopair fuzzy Hamacher weightingaverage (DHq-ROFHWA) operators.

Definition 8. Assume that d j =(h j, g j

)( j = 1, 2, . . . , n) is a list of DHq-ROFNs with weighting vector be

wi = (w1, w2, . . . , wn)T, which satisfies wi ∈ [0, 1] and

∑ni=1 wi = 1. Then the DHq-ROFHWA aggregation

operator can be denoted as

DHq-ROFHWA(d1, d2, · · · , dn) =n⊕

j=1w jd j (10)

According to the operation laws of DHq-ROFNs, we can obtain the computed result:

Theorem 1. The computing results by utilizing DHq-ROFHWA operator is

DHq-ROFHWA(d1, d2, · · · , dn) =n⊕

j=1w jd j

= ∪α j∈h j,β j∈g j

q

√√√√√√√ n∏j=1

(1+(γ−1)(α j)

q)wj−

n∏j=1

(1−(α j)

q)wj

n∏j=1

(1+(γ−1)(α j)

q)wj+(γ−1)n∏

j=1

(1−(α j)

q)wj

,

q√γ

n∏j=1(β j)

wj

q

√n∏

j=1

(1+(γ−1)

(1−(β j)

q))wj+(γ−1)n∏

j=1(β j)

qwj

(11)

Symmetry 2019, 11, 771 7 of 26

Example 1. Given four dual hesitant q-rung orthopair fuzzy numbers: d1 = {{0.7, 0.8}, {0.5}}, d2 =

{{0.4}, {0.6}}, d3 = {{0.6}, {0.7, 0.9}}, d4 = {{0.3}, {0.2}} with weighting vector be w j = (0.4, 0.1, 0.3, 0.2),suppose that q = 3,γ = 3, then for membership degree α, we can derive

α1 = DHq-ROFHWA(0.7, 0.4, 0.6, 0.3)

= 3

√√√√√√√√√√√√√√√√√√√√√√√√√√

((

1 + 2× 0.73)0.4×

(1 + 2× 0.43

)0.1×

(1 + 2× 0.63

)0.3×

(1 + 2× 0.33

)0.2)

−

((1− 0.73

)0.4×

(1− 0.43

)0.1×

(1− 0.63

)0.3×

(1− 0.33

)0.2)

((1 + 2× 0.73

)0.4×

(1 + 2× 0.43

)0.1×

(1 + 2× 0.63

)0.3×

(1 + 2× 0.33

)0.2)

+2×((

1− 0.73)0.4×

(1− 0.43

)0.1×

(1− 0.63

)0.3×

(1− 0.33

)0.2)

= 0.5284

In the same way, we have α2 = DHq-ROFHWA(0.8, 0.4, 0.6, 0.3) = 0.5787, thus α = {0.5284, 0.5787}.For nonmembership β, we can derive

β1 = DHq-ROFHWA(0.5, 0.6, 0.7, 0.2)

=

3√3×0.50.4

×0.60.1×0.70.3

×0.20.2

q

√√√√√√√√√ (1 + 2×

(1− 0.53

))0.4×

(1 + 2×

(1− 0.63

))0.1×

(1 + 2×

(1− 0.73

))0.3

×

(1 + 2×

(1− 0.23

))0.2+ 2× 0.51.2

× 0.60.3× 0.70.9

× 0.20.6

= 0.2498

In the same way, we have β2 = DHq-ROFHWA(0.5, 0.6, 0.9, 0.2) = 0.2668, thus β = {0.2498, 0.2668}.So DHq-ROFHWA(d1, d2, d3, d4) = {{0.5284, 0.5787} , {0.2498, 0.2668}}.

It’s clear that the DHq-ROFHWA operator satisfies some properties including Idempotency,Monotonicity and Boundedness.

Property 1. (Idempotency) Assume that d j =(h j, g j

)( j = 1, 2, . . . , n) are equal, we can obtain

DHq-ROFHWA(d1, d2, . . . , dn) = d (12)

Property 2. (Monotonicity) Let d j =(h j, g j

)and d′ j =

(h′j, g′j

), j = 1, 2, . . . , n be two sets of DHq-ROFNs.

If h j ≤ h′j and g j ≥ g′j hold for all j, then

DHq-ROFHWA(d1, d2, . . . , dn) ≤ DHq-ROFHWA(d′1, d′2, . . . , d′n) (13)

Property 3. (Boundedness) Assume that d j =(h j, g j

)( j = 1, 2, . . . , n) be a set of DHq-ROFNs. If d+ =

∪α j∈h j,β j∈g j

{{maxi(αi)

},{mini(βi)

}}, d− = ∪α j∈h j,β j∈g j

{{mini(αi)

},{maxi(βi)

}}, then

d− ≤ DHq-ROFHWA(d1, d2, . . . , dn) ≤ d+ (14)

Next, by changing γ and q we shall derive some special results.

Symmetry 2019, 11, 771 8 of 26

Case 1. When γ = 1, the DHq-ROFHWA operator is going to degrade into the dual hesitant q-rung orthopairfuzzy weighting average (DHq-ROFWA) aggregation operator presented as

DHq-ROFWA(d1, d2, · · · , dn) =n⊕

j=1w jd j

= ∪α j∈h j,β j∈g j

q

√1−

n∏j=1

(1−

(α j

)q)w j

,

n∏j=1

(β j

)w j

(15)

Case 2. When γ = 2, the DHq-ROFHWA operator is going to degrade into the dual hesitant q-rung orthopairfuzzy Einstein weighting average (DHq-ROEWA) operator, presented as

DHq-ROFEWA(d1, d2, · · · , dn) =n⊕

j=1w jd j

= ∪α j∈h j,β j∈g j

q

√√√√√√√ n∏j=1

(1+(α j)

q)wj−

n∏j=1

(1−(α j)

q)wj

n∏j=1

(1+(α j)

q)wj+n∏

j=1

(1−(α j)

q)wj

,

q√2

n∏j=1(β j)

wj

q

√n∏

j=1

(2−(β j)

q)wj+n∏

j=1(β j)

qwj

(16)

Case 3. When q = 1, the DHq-ROFHWA operator is going to degrade into the dual hesitant intuitionistic fuzzyHamacher weighting average (DHIFHWA) operator, presented as

DHIFHWA(d1, d2, · · · , dn) =n⊕

j=1w jd j

= ∪α j∈h j,β j∈g j

n∏j=1

(1+(γ−1)(α j))wj−

n∏j=1(1−(α j))

wj

n∏j=1

(1+(γ−1)(α j))wj+(γ−1)

n∏j=1(1−(α j))

wj

,γ

n∏j=1(β j)

wj

n∏j=1(1+(γ−1)(1−(β j)))

wj+(γ−1)n∏

j=1(β j)

wj

(17)

Case 4. When q = 2, the DHq-ROFHWA operator is going to degrade into the dual hesitant Pythagorean fuzzyHamacher weighting average (DHPFHWA) operator [56], presented as

DHPFHWA(d1, d2, · · · , dn) =n⊕

j=1w jd j

= ∪α j∈h j,β j∈g j

√√√√√√√ n∏

j=1

(1+(γ−1)(α j)

2)wj−

n∏j=1

(1−(α j)

2)wj

n∏j=1

(1+(γ−1)(α j)

2)wj

+(γ−1)n∏

j=1

(1−(α j)

2)wj

,

√γ

n∏j=1(β j)

wj

√n∏

j=1

(1+(γ−1)

(1−(β j)

2))wj

+(γ−1)n∏

j=1(β j)

2wj

(18)

Furthermore, to consider the order positions of being fused arguments, we develop the dualhesitant q-rung orthopair fuzzy Hamacher ordered weighting average (DHq-ROFHOWA) operatoras follows.

Symmetry 2019, 11, 771 9 of 26

Definition 9. Assume that d j =(h j, g j

)( j = 1, 2, . . . , n) is a group of DHq-ROFNs, the dual hesitant q-rung

orthopair fuzzy Hamacher ordered weighting average (DHq-ROFHOWA) operator with associated weighting

vector be w = (w1, w2, · · · , wn)T, which satisfies the conditions of w j > 0 and

n∑j=1

w j = 1, then

DHq-ROFHOWA(d1, d2, · · · , dn) =n⊕

j=1w jdσ( j)

= ∪ασ( j)∈h j,βσ( j)∈g j

q

√√√√√√√ n∏j=1

(1+(γ−1)(ασ( j))

q)wj−

n∏j=1

(1−(ασ( j))

q)wj

n∏j=1

(1+(γ−1)(ασ( j))

q)wj+(γ−1)n∏

j=1

(1−(ασ( j))

q)wj

,

q√γ

n∏j=1(βσ( j))

wj

q

√n∏

j=1

(1+(γ−1)

(1−(βσ( j))

q))wj+(γ−1)n∏

j=1(βσ( j))

qwj

(19)

where (σ(1), σ(2), · · · , σ(n)) is a permutation of (1, 2, · · · , n), and dσ( j−1) ≥ dσ( j) for all j = 2, · · · , n.

It is clear that the DHq-ROFHOWA operator satisfies some properties including idempotency,monotonicity, and boundedness.

Property 4. (Idempotency) Assume that d j =(h j, g j

)( j = 1, 2, . . . , n) are equal, we can obtain

DHq-ROFHOWA(d1, d2, . . . , dn) = d (20)

Property 5. (Monotonicity) Assume that d j =(h j, g j

)and d′j =

(h′j, g′j

), j = 1, 2, . . . , n are two groups of

DHq-ROFNs. If h j ≤ h′j and g j ≥ g′j hold for all j, then

DHq-ROFHOWA(d1, d2, . . . , dn) ≤ DHq-ROFHOWA(d′1, d′3, . . . , d′n) (21)

Property 6. (Boundedness) Assume that d j =(h j, g j

)( j = 1, 2, . . . , n) is a set of DHq-ROFNs. If d+ =

∪ασ( j)∈h j,βσ( j)∈g j

{{maxi(αi)

},{mini(βi)

}}, d− = ∪ασ( j)∈h j,βσ( j)∈g j

{{mini(αi)

},{maxi(βi)

}}, then

d− ≤ DHq-ROFHOWA(d1, d2, . . . , dn) ≤ d+ (22)

Next, by changing γ and q we shall derive some special results.

Case 1. When γ = 1, the DHq-ROFHOWA operator is going to degrade into the dual hesitant q-rung orthopairfuzzy ordered weighting average (DHq-ROFOWA) operator, presented as

DHq-ROFOWA(d1, d2, · · · , dn) =n⊕

j=1w jdσ( j)

= ∪ασ( j)∈h j,βσ( j)∈g j

q

√1−

n∏j=1

(1−

(ασ( j)

)q)w j

,

n∏j=1

(βσ( j)

)w j

(23)

Symmetry 2019, 11, 771 10 of 26

Case 2. When γ = 2, the DHq-ROFHOWA operator is going to degrade into the dual hesitant q-rung orthopairfuzzy Einstein ordered weighting average (DHq-ROEOWA) operator, presented as

DHq-ROFEOWA(d1, d2, · · · , dn) =n⊕

j=1w jdσ( j)

= ∪ασ( j)∈h j,β j∈gσ( j)

q

√√√√√√√ n∏j=1

(1+(ασ( j))

q)wj−

n∏j=1

(1−(ασ( j))

q)wj

n∏j=1

(1+(ασ( j))

q)wj+n∏

j=1

(1−(ασ( j))

q)wj

,

q√2

n∏j=1(βσ( j))

wj

q

√n∏

j=1

(2−(βσ( j))

q)wj+n∏

j=1(βσ( j))

qwj

(24)

Case 3. When q = 1, the DHq-ROFHOWA operator is going to degrade into the dual hesitant intuitionisticfuzzy Hamacher ordered weighting average (DHIFHOWA) operator, presented as

DHIFHOWA(d1, d2, · · · , dn) =n⊕

j=1w jdσ( j)

= ∪α j∈hσ( j),β j∈gσ( j)

n∏j=1

(1+(γ−1)(ασ( j)))wj−

n∏j=1(1−(ασ( j)))

wj

n∏j=1

(1+(γ−1)(ασ( j)))wj+(γ−1)

n∏j=1(1−(ασ( j)))

wj

,γ

n∏j=1(βσ( j))

wj

n∏j=1(1+(γ−1)(1−(βσ( j))))

wj+(γ−1)n∏

j=1(βσ( j))

wj

(25)

Case 4. When q = 2, the DHq-ROFHOWA operator is going to degrade into the dual hesitant Pythagoreanfuzzy Hamacher ordered weighting average (DHPFHOWA) operator [56], presented as

DHPFHOWA(d1, d2, · · · , dn) =n⊕

j=1w jdσ( j)

= ∪ασ( j)∈h j,βσ( j)∈g j

√√√√√√√ n∏

j=1

(1+(γ−1)(ασ( j))

2)wj−

n∏j=1

(1−(ασ( j))

2)wj

n∏j=1

(1+(γ−1)(ασ( j))

2)wj

+(γ−1)n∏

j=1

(1−(ασ( j))

2)wj

,

√γ

n∏j=1(βσ( j))

wj

√n∏

j=1

(1+(γ−1)

(1−(βσ( j))

2))wj

+(γ−1)n∏

j=1(βσ( j))

2wj

(26)

According to Definitions 8–9, we can obtain that the DHq-ROFHWA operators can only weigh theDHq-ROFN itself, while the DHq-ROFHOWA operators can only weigh the ordered positions of theDHq-ROFN. To consider both two weights, we will develop the dual hesitant q-rung orthopair fuzzyHamacher hybrid averaging (DHq-ROFHHA) operator as follows.

Symmetry 2019, 11, 771 11 of 26

Definition 10. Let d j =(h j, g j

)( j = 1, 2, . . . , n) be a group of DHq-ROFNs. A dual hesitant q-rung orthopair

fuzzy Hamacher hybrid average (DHq-ROFHHA) operator mapping DHq-ROFHHA: Pn→ P, such that

DHq-ROFHHA(d1, d2, · · · , dn) =n⊕

j=1w j

.dσ( j)

= ∪ .ασ( j)∈h j,

.βσ( j)∈g j

q

√√√√√√√ n∏j=1

(1+(γ−1)(

.ασ( j))

q)wj−

n∏j=1

(1−(

.ασ( j))

q)wj

n∏j=1

(1+(γ−1)(

.ασ( j))

q)wj+(γ−1)n∏

j=1

(1−(

.ασ( j))

q)wj

,

q√γ

n∏j=1

( .βσ( j)

)wj

q

√n∏

j=1

(1+(γ−1)

(1−

( .βσ( j)

)q))wj+(γ−1)

n∏j=1

( .βσ( j)

)qwj

(27)

where w = (w1, w2, · · · , wn) means the associated weights, which satisfies w j ∈ [0, 1],n∑

j=1w j = 1, and

.dσ( j)

denotes the j-th largest number of DHq-ROFNs.d j

( .d j =

(nω j

)d j, j = 1, 2, · · · , n

), ω = (ω1,ω2, · · · ,ωn)

represents the weights of the DHq-ROFNs d j( j = 1, 2, · · · , n), which satisfies ω j ∈ [0, 1],n∑

j=1ω j = 1, and n

indicates the balance coefficient.When w = (1/n, 1/n, · · · , 1/n)T, the DHq-ROFHHA operator is going to degrade into the

DHq-ROFHWA operator; when ω = (1/n, 1/n, · · · , 1/n), the DHq-ROFHHA operator is going to degradeinto the dual (DHq-ROFHOWA operator.

Next, by changing γ and q, we shall derive some special results.

Case 1. When γ = 1, theDHq-ROFHHA operator is going to degrade into the dual hesitant q-rung orthopairfuzzy hybrid averaging (DHq-ROFHA) operator, presented as

DHq-ROFHA(d1, d2, · · · , dn) =n⊕

j=1w j

.dσ( j)

= ∪ .ασ( j)∈h j,

.βσ( j)∈g j

q

√1−

n∏j=1

(1−

( .ασ( j)

)q)w j

,

n∏j=1

( .βσ( j)

)w j

(28)

Case 2. When γ = 2, the DHq-ROFHHA operator is going to degrade into the dual hesitant q-rung orthopairfuzzy Einstein hybrid averaging (DHq-ROFEHA) operator, presented as

DHq-ROFEHA(d1, d2, · · · , dn) =n⊕

j=1w j

.dσ( j)

= ∪ .ασ( j)∈h j,

.β j∈gσ( j)

q

√√√√√√√ n∏j=1

(1+(

.ασ( j))

q)wj−

n∏j=1

(1−(

.ασ( j))

q)wj

n∏j=1

(1+(

.ασ( j))

q)wj+n∏

j=1

(1−(

.ασ( j))

q)wj

,

q√2

n∏j=1

( .βσ( j)

)wj

q

√n∏

j=1

(2−

( .βσ( j)

)q)wj+

n∏j=1

( .βσ( j)

)qwj

(29)

Symmetry 2019, 11, 771 12 of 26

Case 3. When q = 1, the DHq-ROFHHA operator is going to degrade into the dual hesitant intuitionistic fuzzyHamacher hybrid averaging (DHIFHHA) operator, presented as

DHIFHHA(d1, d2, · · · , dn) =n⊕

j=1w j

.dσ( j)

= ∪ .α j∈hσ( j),

.β j∈gσ( j)

n∏j=1

(1+(γ−1)(.ασ( j)))

wj−

n∏j=1(1−(

.ασ( j)))

wj

n∏j=1

(1+(γ−1)(.ασ( j)))

wj+(γ−1)n∏

j=1(1−(

.ασ( j)))

wj

,γ

n∏j=1

( .βσ( j)

)wj

n∏j=1

(1+(γ−1)

(1−

( .βσ( j)

)))wj+(γ−1)n∏

j=1

( .βσ( j)

)wj

(30)

Case 4. When q = 2, the DHq-ROFHHA operator is going to degrade into the dual hesitant Pythagorean fuzzyHamacher hybrid averaging (DHPFHHA) operator [56]:

DHPFHHA(d1, d2, · · · , dn) =n⊕

j=1w j

.dσ( j)

= ∪ .ασ( j)∈h j,

.βσ( j)∈g j

√√√√√√√ n∏

j=1

(1+(γ−1)(

.ασ( j))

2)wj−

n∏j=1

(1−(

.ασ( j))

2)wj

n∏j=1

(1+(γ−1)(

.ασ( j))

2)wj

+(γ−1)n∏

j=1

(1−(

.ασ( j))

2)wj

,

√γ

n∏j=1

( .β j

)wj

√n∏

j=1

(1+(γ−1)

(1−

( .βσ( j)

)2))wj+(γ−1)

n∏j=1

( .βσ( j)

)2wj

(31)

4.2. Dual Hesitant q-Rung Orthopair Fuzzy Hamacher Geometric Operators

In accordance with the DHq-ROFHWA aggregation operators and the geometric operations,we will define some dual hesitant q-rung orthopair fuzzy Hamacher weighting geometric(DHq-ROFHWG) aggregation operators as follows.

Definition 11. Let d j =(h j, g j

)( j = 1, 2, . . . , n) be a collection of DHq-ROFNs. The dual hesitant q-rung

orthopair fuzzy Hamacher weighting geometric (DHq-ROFHWG) aggregation operator can be depicted as

DHq-ROFHWG(d1, d2, · · · , dn) =n⊗

j=1

(d j

)w j (32)

According to Hamacher operations of DHq-ROFNs, we can obtain the computed result as follows:

Theorem 2. The fused results by utilizing DHq-ROFHWA operator can be shown as

DHq-ROFHWG(d1, d2, · · · , dn) =n⊗

j=1

(d j

)w j

= ∪α j∈h j,β j∈g j

q√γn∏

j=1(α j)

wj

q

√n∏

j=1

(1+(γ−1)

(1−(α j)

q))wj+(γ−1)n∏

j=1(α j)

qwj

,

q

√√√√√√√ n∏j=1

(1+(γ−1)(β j)

q)wj−

n∏j=1

(1−(β j)

q)wj

n∏j=1

(1+(γ−1)(β j)

q)wj+(γ−1)n∏

j=1

(1−(β j)

q)wj

(33)

Symmetry 2019, 11, 771 13 of 26

Example 2. Given four dual hesitant q-rung orthopair fuzzy numbers—d1 = {{0.7, 0.8}, {0.5}}, d2 =

{{0.4}, {0.6}}, d3 = {{0.6}, {0.7, 0.9}}, d4 = {{0.3}, {0.2}}—with weighting vector will be w j = (0.4, 0.1, 0.3, 0.2),suppose that q = 3,γ = 3, then for membership degree α, we can derive

α1 = DHq-ROFHWG(0.7, 0.4, 0.6, 0.3)

=

3√3×0.70.4

×0.40.1×0.60.3

×0.30.2

q

√√√√√√√√√ (1 + 2×

(1− 0.73

))0.4×

(1 + 2×

(1− 0.43

))0.1×

(1 + 2×

(1− 0.63

))0.3

×

(1 + 2×

(1− 0.33

))0.2+ 2× 0.71.2

× 0.40.3× 0.60.9

× 0.30.6

= 0.1040

In the same way, we have α2 = DHq-ROFHWG(0.8, 0.4, 0.6, 0.3) = 0.1104, thus, α = {0.1040, 0.1104}.For nonmembership β, we can derive

β1 = DHq-ROFHWG(0.5, 0.6, 0.7, 0.2)

= 3

√√√√√√√√√√√√√√√√√√√√√√√√√√

((

1 + 2× 0.53)0.4×

(1 + 2× 0.63

)0.1×

(1 + 2× 0.73

)0.3×

(1 + 2× 0.23

)0.2)

−

((1− 0.53

)0.4×

(1− 0.63

)0.1×

(1− 0.73

)0.3×

(1− 0.23

)0.2)

((1 + 2× 0.53

)0.4×

(1 + 2× 0.63

)0.1×

(1 + 2× 0.73

)0.3×

(1 + 2× 0.23

)0.2)

+2×((

1− 0.53)0.4×

(1− 0.63

)0.1×

(1− 0.73

)0.3×

(1− 0.23

)0.2)

= 0.5553

In the same way, we have β2 = DHq-ROFHWG(0.5, 0.6, 0.9, 0.2) = 0.6003, thus, β = {0.5553, 0.6003}.So DHq-ROFHWG(d1, d2, d3, d4) = {{0.1040, 0.1104} , {0.5553, 0.6003}}.

It is clear that the DHq-ROFHWG operator satisfies some properties including idempotency,monotonicity, and boundedness.

Property 7. (Idempotency) Assume that d j =(h j, g j

)( j = 1, 2, . . . , n) are equal, we can obtain

DHq-ROFHWG(d1, d2, . . . , dn) = d (34)

Property 8. (Monotonicity) Assume that d j =(h j, g j

)and d′j =

(h′j, g′j

), j = 1, 2, . . . , n be two sets of

DHq-ROFNs. If h j ≤ h′j and g j ≥ g′j hold for all j, then

DHq-ROFHWG(d1, d2, . . . , dn) ≤ DHq-ROFHWG(d′1, d′1, . . . , d′n) (35)

Property 9. (Boundedness) Assume that d j =(h j, g j

)( j = 1, 2, . . . , n) be a set of DHq-ROFNs. If d+ =

∪α j∈h j,β j∈g j

{{maxi(αi)

},{mini(βi)

}}, d− = ∪α j∈h j,β j∈g j

{{mini(αi)

},{maxi(βi)

}}, then

d− ≤ DHq-ROFHWG(d1, d2, . . . , dn) ≤ d+ (36)

Next, by changing γ and q, we shall derive some special results.

Symmetry 2019, 11, 771 14 of 26

Case 1. When γ = 1, the DHq-ROFHWG operator is going to degrade into the dual hesitant q-rung orthopairfuzzy weighting geometric (DHq-ROFWG) operator, presented as

DHq-ROFWG(d1, d2, · · · , dn) =n⊗

j=1

(d j

)w j

= ∪α j∈h j,β j∈g j

n∏

j=1

(α j

)w j

,

q

√1−

n∏j=1

(1−

(β j

)q)w j

(37)

Case 2. When γ = 2, the DHq-ROFHWG operator is going to degrade into the dual hesitant q-rung orthopairfuzzy Einstein weighting geometric (DHq-ROFEWG) operator, presented as

DHq-ROFEWG(d1, d2, · · · , dn) =n⊗

j=1

(d j

)w j

= ∪α j∈h j,β j∈g j

q√2n∏

j=1(α j)

wj

q

√n∏

j=1

(2−(α j)

q)wj+n∏

j=1(α j)

qwj

,

q

√√√√√√√ n∏j=1

(1+(β j)

q)wj−

n∏j=1

(1−(β j)

q)wj

n∏j=1

(1+(β j)

q)wj+n∏

j=1

(1−(β j)

q)wj

(38)

Case 3. When q = 1, the DHq-ROFHWG operator is going to degrade into the dual hesitant intuitionisticfuzzy Hamacher weighting geometric (DHIFHWG) operator, presented as

DHIFHWG(d1, d2, · · · , dn) =n⊗

j=1

(d j

)w j

= ∪α j∈h j,β j∈g j

γn∏

j=1(α j)

wj

n∏j=1(1+(γ−1)(1−(α j)))

wj+(γ−1)n∏

j=1(α j)

wj

,n∏

j=1(1+(γ−1)(β j))

wj−

n∏j=1(1−(β j))

wj

n∏j=1

(1+(γ−1)(β j))wj+(γ−1)

n∏j=1(1−(β j))

wj

(39)

Case 4. When q = 2, the DHq-ROFHWG operator is going to degrade into the dual hesitant Pythagorean fuzzyHamacher weighting geometric (DHPFHWG) operator [56], presented as

DHPFHWG(d1, d2, · · · , dn) =n⊗

j=1

(d j

)w j

= ∪α j∈h j,β j∈g j

√γ

n∏j=1(α j)

wj

√n∏

j=1

(1+(γ−1)

(1−(α j)

2))wj

+(γ−1)n∏

j=1(α j)

2wj

,

√√√√√√√ n∏

j=1

(1+(γ−1)(β j)

2)wj−

n∏j=1

(1−(β j)

2)wj

n∏j=1

(1+(γ−1)(β j)

2)wj

+(γ−1)n∏

j=1

(1−(β j)

2)wj

(40)

Furthermore, to consider the order positions of being fused arguments, we proposed the dualhesitant q-rung orthopair fuzzy Hamacher ordered weighted geometric (DHq-ROFHOWG) operatoras follows.

Symmetry 2019, 11, 771 15 of 26

Definition 12. Let d j =(h j, g j

)( j = 1, 2, . . . , n) be a group of DHq-ROFNs, the DHq-ROFHOWG operator

with associated weights w = (w1, w2, · · · , wn)T, which satisfies the condition of w j > 0 and

n∑j=1

w j = 1, then

DHq-ROFHOWG(d1, d2, · · · , dn) =n⊗

j=1

(dσ( j)

)w j

= ∪ασ( j)∈h j,βσ( j)∈g j

q√γn∏

j=1(ασ( j))

wj

q

√n∏

j=1

(1+(γ−1)

(1−(ασ( j))

q))wj+(γ−1)n∏

j=1(ασ( j))

qwj

,

q

√√√√√√√ n∏j=1

(1+(γ−1)(βσ( j))

q)wj−

n∏j=1

(1−(βσ( j))

q)wj

n∏j=1

(1+(γ−1)(βσ( j))

q)wj+(γ−1)n∏

j=1

(1−(βσ( j))

q)wj

(41)

where (σ(1), σ(2), · · · , σ(n)) is a permutation of (1, 2, · · · , n), such that dσ( j−1) ≥ dσ( j) for all j = 2, · · · , n.

We can easily obtain that the DHq-ROFHOWG operator satisfies the following properties.

Property 10. (Idempotency) Assume that d j =(h j, g j

)( j = 1, 2, . . . , n) are equal, we can obtain

DHq-ROFHOWG(d1, d2, . . . , dn) = d (42)

Property 11. (Monotonicity) Assume that d j =(h j, g j

)and d′j =

(h′j, g′j

), j = 1, 2, . . . , n be two sets of

DHq-ROFNs. If h j ≤ h′j and g j ≥ g′j hold for all j, then

DHq-ROFHOWG(d1, d2, . . . , dn) ≤ DHq-ROFHOWG(d′1, d′2, . . . , d′n) (43)

Property 12. (Boundedness) Assume that d j =(h j, g j

)( j = 1, 2, . . . , n) is a set of DHq-ROFNs. If d+ =

∪ασ( j)∈h j,βσ( j)∈g j

{{maxi(αi)

},{mini(βi)

}}, d− = ∪ασ( j)∈h j,βσ( j)∈g j

{{mini(αi)

},{maxi(βi)

}}, then

d− ≤ DHq-ROFHOWG(d1, d2, . . . , dn) ≤ d+ (44)

Next, by changing γ and q, we shall derive some special results.

Case 1. When γ = 1, the DHq-ROFHOWG operator is going to degrade into the dual hesitant q-rung orthopairfuzzy ordered weighting geometric (DHq-ROFOWG) operator, presented as

DHq-ROFOWG(d1, d2, · · · , dn) =n⊗

j=1

(dσ( j)

)w j

= ∪ασ( j)∈h j,βσ( j)∈g j

n∏

j=1

(ασ( j)

)w j

,

q

√1−

n∏j=1

(1−

(βσ( j)

)q)w j

(45)

Symmetry 2019, 11, 771 16 of 26

Case 2. When γ = 2, the DHq-ROFHOWG operator is going to degrade into the dual hesitant q-rung orthopairfuzzy Einstein ordered weighting geometric (DHq-ROEOWG) operator, presented as

DHq-ROFEOWG(d1, d2, · · · , dn) =n⊗

j=1

(dσ( j)

)w j

= ∪ασ( j)∈h j,β j∈gσ( j)

q√2n∏

j=1(ασ( j))

wj

q

√n∏

j=1

(2−(ασ( j))

q)wj+n∏

j=1(ασ( j))

qwj

,

q

√√√√√√√ n∏j=1

(1+(βσ( j))

q)wj−

n∏j=1

(1−(βσ( j))

q)wj

n∏j=1

(1+(βσ( j))

q)wj+n∏

j=1

(1−(βσ( j))

q)wj

(46)

Case 3. When q = 1, the DHq-ROFHOWG operator is going to degrade into the dual hesitant intuitionisticfuzzy Hamacher ordered weighting geometric (DHIFHOWG) operator, presented as

DHIFHOWG(d1, d2, · · · , dn) =n⊗

j=1

(dσ( j)

)w j

= ∪α j∈hσ( j),β j∈gσ( j)

γn∏

j=1(ασ( j))

wj

n∏j=1(1+(γ−1)(1−(ασ( j))))

wj+(γ−1)n∏

j=1(ασ( j))

wj

,n∏

j=1(1+(γ−1)(βσ( j)))

wj−

n∏j=1(1−(βσ( j)))

wj

n∏j=1

(1+(γ−1)(βσ( j)))wj+(γ−1)

n∏j=1(1−(βσ( j)))

wj

(47)

Case 4. When q = 2, the DHq-ROFHOWG operator is going to degrade into the dual hesitant Pythagoreanfuzzy Hamacher ordered weighting geometric (DHPFHOWG) operator [56], presented as

DHPFHOWG(d1, d2, · · · , dn) =n⊗

j=1

(dσ( j)

)w j

= ∪ασ( j)∈h j,βσ( j)∈g j

√γ

n∏j=1(ασ( j))

wj

√n∏

j=1

(1+(γ−1)

(1−(ασ( j))

2))wj

+(γ−1)n∏

j=1(ασ( j))

2wj

,

√√√√√√√ n∏

j=1

(1+(γ−1)(βσ( j))

2)wj−

n∏j=1

(1−(βσ( j))

2)wj

n∏j=1

(1+(γ−1)(βσ( j))

2)wj

+(γ−1)n∏

j=1

(1−(βσ( j))

2)wj

(48)

According to Definitions 11 and 12, we can deduce that the DHq-ROFHWG operators can onlyweigh the DHq-ROFN itself, and the DHq-ROFHOWG operators can only weigh the ordered positionsof the DHq-ROFN. In order to consider both two weights, we will develop the dual hesitant q-rungorthopair fuzzy Hamacher hybrid geometric (DHq-ROFHHG) operator as follows.

Symmetry 2019, 11, 771 17 of 26

Definition 13. Let d j =(h j, g j

)( j = 1, 2, . . . , n) be a group of DHq-ROFNs. A dual hesitant q-rung orthopair

fuzzy Hamacher hybrid geometric (DHq-ROFHHG) operator a mapping DHq-ROFHHG: Pn→ P, such that

DHq-ROFHHG(d1, d2, · · · , dn) =n⊗

j=1

( .dσ( j)

)w j

= ∪ .ασ( j)∈h j,

.βσ( j)∈g j

q√γn∏

j=1(

.ασ( j))

wj

q

√n∏

j=1

(1+(γ−1)

(1−(

.ασ( j))

q))wj+(γ−1)n∏

j=1(

.ασ( j))

qwj

,

q

√√√√√√√ n∏j=1

(1+(γ−1)

( .βσ( j)

)q)wj−

n∏j=1

(1−

( .βσ( j)

)q)wj

n∏j=1

(1+(γ−1)

( .βσ( j)

)q)wj+(γ−1)

n∏j=1

(1−

( .βσ( j)

)q)wj

(49)

where w = (w1, w2, · · · , wn) means the associated weights, which satisfies w j ∈ [0, 1],n∑

j=1w j = 1, and

.dσ( j)

is the j-th largest number of DHq-ROFNs.d j

( .d j =

(d j

)nω j , j = 1, 2, · · · , n), ω = (ω1,ω2, · · · ,ωn) means the

weights of DHq-ROFNs d j( j = 1, 2, · · · , n) itself, which satisfies ω j ∈ [0, 1],n∑

j=1ω j = 1, and n indicates the

balance coefficient.When w = (1/n, 1/n, · · · , 1/n)T, the DHq-ROFHHG operator is going to degrade into the

DHq-ROFHWG operator; when ω = (1/n, 1/n, · · · , 1/n), the DHq-ROFHHG operator is going to degradeinto the dual (DHq-ROFHOWG operator.

Next, by changing γ and q, we shall derive some special results.

Case 1. When γ = 1, the DHq-ROFHHG operator is going to degrade into the dual hesitant q-rung orthopairfuzzy hybrid geometric (DHq-ROFHG) operator, presented as

DHq-ROFHG(d1, d2, · · · , dn) =n⊗

j=1

( .dσ( j)

)w j

= ∪ .ασ( j)∈h j,

.βσ( j)∈g j

n∏

j=1

( .ασ( j)

)w j

q

√1−

n∏j=1

(1−

( .βσ( j)

)q)w j

(50)

Case 2. When γ = 2, the DHq-ROFHHG operator is going to degrade into the dual hesitant q-rung orthopairfuzzy Einstein hybrid geometric (DHq-ROFEHG) operator, presented as

DHq-ROFEHG(d1, d2, · · · , dn) =n⊗

j=1

( .dσ( j)

)w j

= ∪ .ασ( j)∈h j,

.β j∈gσ( j)

q√2n∏

j=1(

.ασ( j))

wj

q

√n∏

j=1

(2−(

.ασ( j))

q)wj+n∏

j=1(

.ασ( j))

qwj

,

q

√√√√√√√ n∏j=1

(1+

( .βσ( j)

)q)wj−

n∏j=1

(1−

( .βσ( j)

)q)wj

n∏j=1

(1+

( .βσ( j)

)q)wj+

n∏j=1

(1−

( .βσ( j)

)q)wj

(51)

Symmetry 2019, 11, 771 18 of 26

Case 3. When q = 1, the DHq-ROFHHG operator is going to degrade into the dual hesitant intuitionistic fuzzyHamacher hybrid geometric (DHIFHHG) operator, presented as

DHIFHHG(d1, d2, · · · , dn) =n⊗

j=1

( .dσ( j)

)w j

= ∪ .α j∈hσ( j),

.β j∈gσ( j)

γn∏

j=1(

.ασ( j))

wj

n∏j=1(1+(γ−1)(1−(

.ασ( j))))

wj+(γ−1)n∏

j=1(

.ασ( j))

wj

,n∏

j=1

(1+(γ−1)

( .βσ( j)

))wj−

n∏j=1

(1−

( .βσ( j)

))wj

n∏j=1

(1+(γ−1)

( .βσ( j)

))wj+(γ−1)n∏

j=1

(1−

( .βσ( j)

))wj

(52)

Case 4. When q = 2, the DHq-ROFHHG operator is going to degrade into the dual hesitant Pythagorean fuzzyHamacher hybrid geometric (DHPFHHG) operator [56], presented as

DHPFHHG(d1, d2, · · · , dn) =n⊗

j=1

( .dσ( j)

)w j

= ∪ .ασ( j)∈h j,

.βσ( j)∈g j

√γ

n∏j=1(

.α j)

wj

√n∏

j=1

(1+(γ−1)

(1−(

.ασ( j))

2))wj

+(γ−1)n∏

j=1(

.ασ( j))

2wj

,

√√√√√√√ n∏

j=1

(1+(γ−1)

( .βσ( j)

)2)wj−

n∏j=1

(1−

( .βσ( j)

)2)wj

n∏j=1

(1+(γ−1)

( .βσ( j)

)2)wj+(γ−1)

n∏j=1

(1−

( .βσ( j)

)2)wj

(53)

Given an numerical example with DHq-ROFNs information to briefly depict the decision steps,suppose there are m alternatives Ai denoted by n attributes G j, let w j be attribute weights with0 ≤ w j ≤ 1,

∑nj=1 w j = 1, then the decision-making steps are listed as follows.

Step 1. Collect the dual hesitant q-rung orthopair fuzzy decision-making information given by experts

and construct the evaluation matrix Ri j =(ri j

)m×n

;

Step 2. According to the attribute weights, we can fuse the dual hesitant q-rung orthopair fuzzyinformation by utilizing the equation (11) or (33);Step 3. Compute the score and accuracy results to determine the rank of all the alternatives.

5. Numerical Example and Comparative Analysis

5.1. Numerical Example

Since the prefabricated building has obvious advantages such as high construction quality,environmental protection, and labor-saving compared with the traditional cast-in-place constructionmethod, this construction method has been gradually promoted in the construction field. However,this immature construction method often overlaps with construction safety risks such as on-siteassembly and parallel construction. In addition, the quality of on-site construction personnel isgenerally low, and it is difficult to adapt to the needs of new construction techniques, which is verylikely to cause construction safety accidents. In order to control the incidence of construction-typeconstruction safety accidents with a gradual upward trend, only correct and scientific decision-makingof a construction safety program can be of great significance for the control of PC construction safety.Thus, how to select the scheme of construction project is an interesting topic. To select the schemeof construction project is a classical MADM problem [67,68]. In this part, we shall give an actualapplication about scheme selection of construction project with dual hesitant q-rung orthopair fuzzy

Symmetry 2019, 11, 771 19 of 26

information in order to demonstrate the aggregation operators developed in our manuscript. There arefive possible construction projects Ai(i = 1, 2, 3, 4, 5) to be selected. The experts selects four attribute toestimate the five possible construction projects: 1O G1 is the Human and management factors; 2O G2 isthe Hoisting construction operation factors; 3O G3 is the PC component installation factor; and 4O G4 isthe environmental factor. The five possible construction projects Ai(i = 1, 2, 3, 4, 5) are to be evaluatedusing the dual hesitant q-rung orthopair fuzzy information which is shown in Table 2. (The attribute’sweights are ω = (0.26, 0.42, 0.18, 0.14)T).

Table 2. DHq-ROFN decision matrix (R).

Alternatives G1 G2 G3 G4

A1 {{0.3,0.4},{0.6}} {{0.4,0.5},{0.2,0.3)}} {{0.5,0.6},{0.8}} {{0.1,0.5},{0.7}}A2 {{0.2},{0.4}} {{0.1,0.2,0.3},{0.2}} {{0.5},{0.2,0.3,0.6}} {{0.8},{0.1,0.2}}A3 {{0.7,0.9},{0.1}} {{0.6},{0.3,0.5}} {{0.4,0.5,0.6},{0.1}} {{0.5,0.6,0.7},{0.2}}A4 {{0.4},{0.2})} {{0.3,0.4,0.5},{0.4}} {{0.3,0.5},{0.4}} {{0.4},{0.4,0.5,0.6}}A5 {{0.3,0.4},{0.2}} {{0.4,0.5,0.6},{0.4}} {{0.5,0.6},{0.7}} {{0.2,0.4,0.5},{0.5}}

In the following, we utilize the DHq-ROFHWA operator and the DHq-ROFHWG operator tostudy scheme selection of construction project from five possible construction projects.

Step 1. Based on the decision-making information given in the Table 2, We shall utilize theDHq-ROFHWA operator to derive the overall preference values ri of the construction projectsAi(i = 1, 2, 3, 4, 5) (let γ = 3, q = 3):

r1 = DHq-ROFHWA(r11, r12, r13, r14) =4⊕

j=1w jr1 j

=

{{0.3743, 0.4118, 0.3974, 0.4314, 0.4331, 0.4625, 0.4510, 0.4785,0.3961, 0.4303, 0.4171, 0.4484, 0.4500, 0.4776, 0.4668, 0.4927

},{

0.1971,0.2337

}}r2 = DHq-ROFHWA(r21, r22, r23, r24) =

4⊕

j=1w jr2 j

= {{0.4420, 0.4480, 0.4635}, {0.1053, 0.1158, 0.1106, 0.1216, 0.1219, 0.1339}}

r3 = DHq-ROFHWA(r31, r32, r33, r34) =4⊕

j=1w jr3 j

=

{

0.6014, 0.6131, 0.6285, 0.6084, 0.6198, 0.6349, 0.6182, 0.6292, 0.6439,0.6981, 0.7066, 0.7181, 0.7032, 0.7116, 0.7229, 0.7104, 0.7187, 0.7297

},

{0.1102, 0.1443}

r4 = DHq-ROFHWA(r41, r42, r43, r44) =

4⊕

j=1w jr4 j

= {{0.3467, 0.3761, 0.3904, 0.4145, 0.4455, 0.4647}, {0.1619, 0.1676, 0.1732}}

r5 = DHq-ROFHWA(r51, r52, r53, r54) =4⊕

j=1w jr5 j

=

0.3767, 0.3946, 0.4118, 0.3996, 0.4157, 0.4314, 0.4349, 0.4488, 0.4625,0.4527, 0.4657, 0.4785, 0.4997, 0.5106, 0.5215, 0.5137, 0.5241, 0.5345,0.3983, 0.4145, 0.4303, 0.4190, 0.4339, 0.4484, 0.4517, 0.4647, 0.4776,0.4684, 0.4806, 0.4927, 0.5129, 0.5233, 0.5337, 0.5263, 0.5362, 0.5462

,

{0.1827}

Step 2. Compute the score values S(ri)(i = 1, 2, · · · , 5) of the overall DHq-ROFNs ri(i = 1, 2, · · · , 5)

S(r1) = 0.5378, S(r2) = 0.5451, S(r3) = 0.6499,S(r4) = 0.5322, S(r5) = 0.5497

Symmetry 2019, 11, 771 20 of 26

Step 3. Determine the ordering of all the construction projects Ai(i = 1, 2, 3, 4, 5) with respect tothe score values S(ri)(i = 1, 2, · · · , 5), then we can derive: A3 � A5 � A2 � A1 � A4, and the bestconstruction project is A3.

Based on the DHq-ROFHWG operator, the decision-making steps can be depicted as.

Step 1. Based on the decision-making information given in the Table 2, We shall utilize theDHq-ROFHWG operator to derive the overall preference values ri of the construction projectsAi(i = 1, 2, 3, 4, 5) (let γ = 3, q = 3):

r1 = DHq-ROFHWG(r11, r12, r13, r14) =4⊗

j=1

(r1 j

)w j

=

{{0.1532, 0.1903, 0.1577, 0.1956, 0.1727, 0.2128, 0.1776, 0.2183,0.1653, 0.2043, 0.1700, 0.2097, 0.1859, 0.2275, 0.1911, 0.2332

},{

0.5506,0.5614

}}r2 = DHq-ROFHWG(r21, r22, r23, r24) =

4⊗

j=1

(r2 j

)w j

= {{0.0997, 0.1382, 0.1670}, {0.2772, 0.2815, 0.2868, 0.2908, 0.3561, 0.3588}}

r3 = DHq-ROFHWG(r31, r32, r33, r34) =4⊗

j=1

(r3 j

)w j

=

{

0.2837, 0.2904, 0.2976, 0.2895, 0.2961, 0.3032, 0.2952, 0.3016, 0.3085,0.3216, 0.3281, 0.3351, 0.3271, 0.3333, 0.3400, 0.3324, 0.3384, 0.3449

},

{0.2432, 0.3919}

r4 = DHq-ROFHWG(r41, r42, r43, r44) =

4⊗

j=1

(r4 j

)w j

= {{0.1622, 0.1730, 0.1860, 0.1978, 0.2081, 0.2206}, {0.3664, 0.3860, 0.4114}}

r5 = DHq-ROFHWG(r51, r52, r53, r54) =4⊗

j=1

(r5 j

)w j

=

0.1678, 0.1841, 0.1903, 0.1725, 0.1892, 0.1956, 0.1885, 0.2061, 0.2128,0.1937, 0.2116, 0.2183, 0.2098, 0.2284, 0.2354, 0.2154, 0.2342, 0.2412,0.1807, 0.1978, 0.2043, 0.1857, 0.2031, 0.2097, 0.2025, 0.2206, 0.2275,0.2079, 0.2263, 0.2332, 0.2246, 0.2435, 0.2505, 0.2304, 0.2493, 0.2564

,

{0.4475}

Step 2. Compute the score values S(ri)(i = 1, 2, · · · , 5) of the overall DHq-ROFNs ri(i = 1, 2, · · · , 5):

S(r1) = 0.4177,S(r2) = 0.4861,S(r3) = 0.4971,S(r4) = 0.4742,S(r5) = 0.4601

Step 3. Determine the ordering of all the construction projects Ai(i = 1, 2, 3, 4, 5) with respect tothe score values S(ri)(i = 1, 2, · · · , 5), then we can derive: A3 � A2 � A4 � A5 � A1, and the bestconstruction project is A3.

5.2. Influence of the Parameter on the Final Result

In order to depict the effects on the ordering results by altering parameters of γ and q in theDHq-ROFHWA (DHq-ROFHWG) operators, all the results are list in Tables 3 and 4.

Symmetry 2019, 11, 771 21 of 26

Table 3. Ordering results by altering γ in the DHq-ROFHWA operator.

Alternatives s(A1) s(A2) s(A3) s(A4) s(A5) Ordering

γ = 1 0.5072 0.5512 0.6523 0.5144 0.5291 A3 � A2 � A5 � A4 � A1γ = 2 0.5271 0.5481 0.6512 0.5271 0.5438 A3 � A2 � A5 � A4 � A1γ = 3 0.5378 0.5451 0.6499 0.5322 0.5497 A3 � A5 � A2 � A1 � A4γ = 4 0.5405 0.5422 0.6482 0.5335 0.5509 A3 � A5 � A2 � A1 � A4γ = 6 0.5414 0.5376 0.6456 0.5340 0.5509 A3 � A5 � A1 � A2 � A4γ = 10 0.5408 0.5321 0.6428 0.5339 0.5500 A3 � A5 � A1 � A4 � A2

Table 4. Ordering results by altering γ in the DHq-ROFHWG operator.

Alternatives s(A1) s(A2) s(A3) s(A4) s(A5) Ordering

γ = 1 0.4344 0.4944 0.6062 0.5019 0.4936 A3 � A4 � A2 � A5 � A1γ = 2 0.4213 0.4886 0.5318 0.4821 0.4695 A3 � A2 � A4 � A5 � A1γ = 3 0.4177 0.4861 0.4971 0.4742 0.4601 A3 � A2 � A4 � A5 � A1γ = 4 0.4191 0.4856 0.4878 0.4724 0.4585 A3 � A2 � A4 � A5 � A1γ = 6 0.4212 0.4856 0.4848 0.4719 0.4585 A2 � A3 � A4 � A5 � A1γ = 10 0.4232 0.4858 0.4838 0.4719 0.4590 A2 � A3 � A4 � A5 � A1

Based on the calculated results listed in Tables 3 and 4, the rank of all alternatives is slightlydifferent with different parameters γ in DHq-ROFHWA and DHq-ROFHWG operators. According tothe comparative analysis, when the parameter γ becomes larger, the fused results by DHq-ROFHWAand DHq-ROFHWG operators become smaller, at the same time, the fused results become more andmore steady.

In order to depict the effects on the ordering results by altering parameters of q in the DHq-ROFHWA(DHq-ROFHWG) operators, all the results are list in Tables 5 and 6.

Table 5. Ordering results by altering q in the DHq-ROFHWA operator.

Alternatives s(A1) s(A2) s(A3) s(A4) s(A5) Ordering

q = 1 0.6104 0.6145 0.7666 0.6187 0.6374 A3 � A5 � A4 � A2 � A1q = 2 0.5702 0.5705 0.7123 0.5680 0.5898 A3 � A5 � A2 � A1 � A4q = 3 0.5378 0.5451 0.6499 0.5322 0.5497 A3 � A5 � A2 � A1 � A4q = 4 0.5194 0.5317 0.6063 0.5145 0.5267 A3 � A2 � A5 � A1 � A4q = 6 0.5050 0.5182 0.5588 0.5029 0.5079 A3 � A2 � A5 � A1 � A4

q = 10 0.5004 0.5073 0.5259 0.5001 0.5008 A3 � A2 � A5 � A1 � A4

Table 6. Ordering results by altering q in the DHq-ROFHWG operator.

Alternatives s(A1) s(A2) s(A3) s(A4) s(A5) Ordering

q = 1 0.3438 0.4319 0.5166 0.4087 0.3982 A3 � A2 � A4 � A5 � A1q = 2 0.3810 0.4682 0.5018 0.4472 0.4320 A3 � A2 � A4 � A5 � A1q = 3 0.4177 0.4861 0.4971 0.4742 0.4601 A3 � A2 � A4 � A5 � A1q = 4 0.4428 0.4938 0.4971 0.4879 0.4767 A3 � A2 � A4 � A5 � A1q = 6 0.4709 0.4985 0.4988 0.4973 0.4912 A3 � A2 � A4 � A5 � A1

q = 10 0.4910 0.4999 0.4999 0.4998 0.4982 A3 � A2 � A4 � A5 � A1

Based on the calculated results listed in Tables 5 and 6, the rank of all alternatives is slightlydifferent with different parameters q in DHq-ROFHWA and DHq-ROFHWG operators. According tothe comparative analysis for DHq-ROFHWA operator, when the parameter q becomes larger, the fusedresults by DHq-ROFHWA operator become smaller, at the same time, the fused results become moreand more steady. For DHq-ROFHWG operator, except for s(A2), when the parameter q becomes larger,the fused results by DHq-ROFHWG operator become larger, at the same time, the fused results becomemore and more steady.

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5.3. Comparative Analysis

In this part, we shall compare our defined dual hesitant q-rung orthopair fuzzy Hamacheraggregation operators with other existing information fusion methods such as the q-ROFWA operatorand the q-ROFWG operator developed by Liu and Wang [57] and the dual hesitant Pythagorean fuzzyHamacher aggregation operators proposed by Xu and Wei [56]. From the below analysis, we caneasily obtain that our proposed methods are more flexible and reasonable in the applications ofMADM problems.

(1) Compared our proposed methods with the information fusion operators presented byLiu and Wang [57], our defined operators are mainly characteristic of the advantages that cantake the interrelationship between the being fused arguments into consideration and scientificallyconsider the human’s hesitance in practical MADM problems, whereas the q-ROFWA andq-ROFWG operators developed by Liu and Wang [57] have the limitation of consideringthe interrelationship between being fused arguments and cannot think about the hesitanceof decision-maker. Thus, it is obvious that our methods are more general to express fuzzyinformation. Our method can conquer the disadvantages of two aggregation operators developedby Liu and Wang [57], because the DHq-ROFHWA and DHq-ROFHWG operators can providesmore effective and flexible information fusion and make it more adequate to deal with MADMproblems in which the attributes are dependent. Based on the above mentioned comparisons andanalysis, the DHq-ROFHWA and DHq-ROFHWG operators we developed are better than thetwo aggregation operators developed by Liu and Wang [57] for fusing the dual hesitant q-rungorthopair fuzzy information. Therefore, the DHq-ROFHWA and DHq-ROFHWG operators aremore valid to handle multiple attribute decision-making under dual hesitant q-rung orthopairfuzzy environment.

(2) Compared our proposed methods with the dual hesitant Pythagorean fuzzy Hamacher operatorspresented by Xu and Wei [56], if we let the parameter q = 2, it is clear that dual hesitantPythagorean fuzzy Hamacher operators presented by Xu and Wei [56] are special cases ofour methods. Evidently, our methods can express more fuzzy information and apply broadlysituations in real MADM problems. Furthermore, in complicated decision-making environment,the decision-maker’s risk attitude is an important factor to think about, our methods can makethis come true by altering the parameter’s q, whereas dual hesitant Pythagorean fuzzy Hamacheroperators presented by Xu and Wei [56] do not have the ability that dynamic adjust to theparameter based on the decision-maker’s risk attitude, thus, it is difficult to deal with the riskmultiple attribute decision-making (MADM) in real practice.

6. Conclusions

Based on the Hamacher operation laws, we utilize the Hamacher weighting average (HWA)operator and Hamacher weighting geometric (HWG) operator to develop some DHq-ROFHWA andDHq-ROFHWG aggregation operators with DHq-ROFNs: the dual hesitant q-rung orthopair fuzzyHamacher weighting average (DHq-ROFHWA) operator, the dual hesitant q-rung orthopair fuzzyHamacher weighting geometric (DHq-ROFHWG) operator, the dual hesitant q-rung orthopair fuzzyHamacher ordered weighted average (DHq-ROFHOWA) operator, the dual hesitant q-rung orthopairfuzzy Hamacher ordered weighting geometric (DHq-ROFHOWG) operator, the dual hesitant q-rungorthopair fuzzy Hamacher hybrid average (DHq-ROFHHA) operator, and the dual hesitant q-rungorthopair fuzzy Hamacher hybrid geometric (DHq-ROFHHG) operator. Of course, the precious meritsand some special cases of these defined operators are investigated. In the end, we take a concreteexample for appraising the construction scheme selection to demonstrate our defined model and totestify its accuracy and scientific. It is clear that our defined operators can consider human’s hesitanceand the interrelationship between being fused arguments, in addition, the newly developed methodsalso can dynamic adjust to the parameter based on the decision-maker’s risk attitude. However,

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the limitation of our approach is that the calculation formula is too complicated, thus, in future we willcontinue to study MADM problems and propose some simplified operators to other decision-makingfields [69–74]. Furthermore, the application of the proposed methods in addressing practical MADMproblems in other manufacturing environment, such as selection of an automated inspection system,selection of an industrial robot, selection of an additive manufacturing process and selection of amachine tool and will also be studied in our future work.

Author Contributions: P.W., G.W., J.W., R.L. and Y.W. conceived and worked together to achieve this work,J.W. compiled the computing program by Excel and analyzed the data, J.W. and G.W. wrote the paper. Finally, allthe authors have read and approved the final manuscript.

Funding: The work was supported by the National Natural Science Foundation of China under Grant No.71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republicof China (14YJCZH091). The APC was funded by National Natural Science Foundation of China under GrantNo. 71571128.

Conflicts of Interest: The authors declare no conflict of interest.

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